Several numerical methods for solution of vibrational Schrödinger equation in the course of hybrid statistical physics – quantum mechanical modeling of molecular solvation phenomena were applied, tested, and compared. The mentioned numerical methods were applied to compute the anharmonic OH stretching vibrational frequencies of the free and solvated hydroxide anion in diluted water solutions on the basis of one-dimensional vibrational potential energies computed at various levels of theory, including density functional theory based methodologies, as well as methods based on many-body perturbation theory. The tested methods included: i) simple Hamiltonian matrix diagonalization technique, based on representation of the vibrational potential in Simons-Parr-Finlan (SPF) coordinates, ii) Numerov algorithm and iii) Fourier grid Hamiltonian method (FGH). Considering the Numerov algorithm as a reference method, the diagonalization technique performs remarkably well in a very wide range of frequencies and frequency shifts. FGH method, on the other hand, though showing a very good performance as well, exhibits more significant (and non-uniform) discrepancies with the Numerov algorithm, even for rather modest frequency shifts. Particular aspects related to HPC-implementation of the numerical algorithms for the applied methodologies were addressed.
Fourier grid Hamiltonian method; Numerov algorithm; diagonalization of Hamiltonian matrix; solvation; intermolecular interactions; anharmonic O-H vibrational frequency shifts; Monte-Carlo simulation